Discrete time and continuous time

In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "points in time", or equivalently as being unchanged throughout each non-zero region of time ("time period")—that is, time is viewed as a discrete variable. Thus a non-time variable jumps from one value to another as time moves from one time period to the next. This view of time corresponds to a digital clock that gives a fixed reading of 10:37 for a while, and then jumps to a new fixed reading of 10:38, etc. In this framework, each variable of interest is measured once at each time period. The number of measurements between any two time periods is finite. Measurements are typically made at sequential integer values of the variable "time". A discrete signal or discrete-time signal is a time series consisting of a sequence of quanti

Essays in Empirical Asset Pricing

Alexis Arilès Marchal

This thesis consists of three applications of machine learning techniques to empirical asset pricing.In the first part, which is co-authored work with Oksana Bashchenko, we develop a new method that detects jumps nonparametrically in financial time series and significantly outperforms the current benchmark on simulated data. We use a long short-term memory (LSTM) neural network that is trained on labelled data generated by a process that experiences both jumps and volatility bursts. As a result, the network learns how to disentangle the two. Then it is applied to out-of-sample simulated data and delivers results that considerably differ from the benchmark: we obtain fewer spurious detection and identify a larger number of true jumps. When applied to real data, our approach for jump screening allows to extract a more precise signal about future volatility.In the second part, which is co-authored work with Oksana Bashchenko, we develop a methodology for detecting asset bubbles using a neural network. We rely on the theory of local martingales in continuous-time and use a deep network to estimate the diffusion coefficient of the price process more accurately than the current estimator, obtaining an improved detection of bubbles. We show the outperformance of our algorithm over the existing statistical method in a laboratory created with simulated data. We then apply the network classification to real data and build a zero net exposure trading strategy that exploits the risky arbitrage emanating from the presence of bubbles in the US equity market from 2006 to 2008. The profitability of the strategy provides an estimation of the economical magnitude of bubbles as well as support for the theoretical assumptions relied on.In the third part, I propose a new tool to characterize the resolution of uncertainty around FOMC press conferences. It relies on the construction of a measure capturing the level of discussion complexity between the Fed Chair and reporters during the Q&A sessions. I show that complex discussions are associated with higher equity returns and a drop in realized volatility. The method creates an attention score by quantifying how much the Chair needs to rely on reading internal documents to be able to answer a question. This is accomplished by building a novel dataset of video images of the press conferences and leveraging recent deep learning algorithms from computer vision. This alternative data provides new information on nonverbal communication that cannot be extracted from the widely analyzed FOMC transcripts. This chapter can be seen as a proof of concept that certain videos contain valuable information for the study of financial markets.

EPFL2022

Graph Laplacians for Rotation Equivariant Convolutional Neural Networks

Martino Milani

A fundamental problem in signal processing is to design computationally efficient algorithms to filter signals. In many applications, the signals to filter lie on a sphere. Meaningful examples of data of this kind are weather data on the Earth, or images of the sky. It is then important to design filtering algorithms that are computationally efficient and capable of exploiting the rotational symmetry of the problem. In these applications, given a continuous signal

f: \mathbb S^2 \rightarrow \mathbb R

on a 2-sphere

\mathbb S^2 \subset \mathbb R^3

, we can only know the vector of its sampled values

\mathbf f \in \mathbb R^N:\ (\mathbf f)_i = f(\mathbf x_i)

in a finite set of points

\mathcal P \subset \mathbb S^2,\quad \mathcal P = \{\mathbf x_i\}_{i=0}^{n-1}

where our sensors are located. Perraudin et al. in \cite{DeepSphere} construct a sparse graph

G

on the vertex set

\mathcal P

and then use a polynomial of the corresponding graph Laplacian matrix

\mathbf L \in \mathbb R^{n\times n}

to perform a computationally efficient -

\mathcal O (n)

- filtering of the sampled signal

\mathbf f

. In order to study how well this algorithm respects the symmetry of the problem - i.e it is equivariant to the rotation group SO(3) - it is important to guarantee that the spectrum of

\mathbf L

and spectrum of the Laplace-Beltrami operator

\Delta_\mathbb S^2

are somewhat ``close''. We study the spectral properties of such graph Laplacian matrix in the special case of \cite{DeepSphere} where the sampling

\mathcal P

is the so called HEALPix sampling (acronym for \textbf Hierarchical \textbf Equal \textbf Area iso\textbf Latitude \textbf {Pix}elization) and we show a way to build a graph

G'

such that the corresponding graph Laplacian matrix

\mathbf L'

shows better spectral properties than the one presented in \cite{DeepSphere}. We investigate other different methods of building the matrix

\mathbf L

better suited to non uniform sampling measures. In particular, we studied the Finite Element Method approximation of the Laplace-Beltrami operator on the sphere, and how FEM filtering relates to graph filtering, showing the importance of non symmetric discrete Laplacians when it comes to non uniform sampling measures. We finish by showing how the graph Laplacian

\mathbf L'

proposed in this work improved the performances of DeepSphere in a well known classification task using different sampling schemes of the sphere, and by comparing the different Discrete Laplacians introduced in this work.

2019

Haptic interface design and control with application to surgery simulation

Ulrich Spälter

This thesis proposes a framework for haptic devices interacting with virtual environments. As application of the theoretical results a haptic device for the surgical training of hysteroscopy, a gynecologic intervention, is presented. Surgery simulators, which can be described as 'flight simulators for surgeons', comprise a virtual reality for modeling and visualization of organs, and the haptic device: While the surgeon interactively follows the scene on the computer screen, he perceives the contact forces at the tool handle of the surgical instrument. The haptic device tracks the position of the surgery tool and transforms the virtual contact forces to real forces, which can be actually felt by the surgeon. This thesis treats haptic devices with application to surgery simulators for minimally-invasive surgery. Haptic interfaces are complex mechatronic systems and show inherent trade-offs between haptic realism and system stability. Human factors play an important role, as the human operator can both stabilize or destabilize the system. The haptic perception, which can vary between individuals, as a criterion for haptic rendering quality is still not formalized. The optimization of haptic interfaces, within focus of research since 15 years ago, has remained a complex task. However, with look at the emerging applications for haptics, it has become an even more important topic. For this reason it is necessary to integrate mechanical design, control design, mechatronic elements and human factors in a unifying approach. This thesis highlights the principal trade-offs of haptic interfaces and works out design guidelines for new developments and quantitative information on where and how to optimize haptic systems. As application example a haptic interface for the simulation of hysteroscopy, a gynecologic intervention on the female uterus, has been realized as prototype. The device takes the characteristics of hysteroscopy into account and introduces features like insertion and complete removal of the surgery tool from the haptic interface. A design methodology is proposed, which combines aspects of theoretical kinematics based on Lie-Groups with integrated product development – a systematic approach for engineering tasks. The result, a novel kinematic design, has been realized as haptic interface prototype. The well-known affine parameterization for all stabilizing controllers is shown to be a well-suited tool for haptic control synthesis. It provides insights into the control trade-offs and allows to integrate human factors at controller design stage. The resulting controller, demonstrated by experimental results, can reduce parasitic effects (friction, device dynamics) close to and below the human perception threshhold, thus making the device transparent within the specified bandwidth and suitable for real-time implementation. For numerical controller representation the δ-operator is discussed. It unifies continuous and discrete time domain and has beneficial numerical properties. As there is still no standardized procedure for technical performance evaluation of haptic interfaces methods proposed in literature are summarized. In the near future a large market for haptic technology can be expected with applications from surgery-assist devices and drive-by-wire interfaces in automobiles to entertainment, which highlights the relevance of the treated topic.

EPFL2006